Process for finding endmembers in a data set
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Abstract
Description
 [0001]This application claims priority of Provisional application Ser. No. 60/655,185, filed on Feb. 22, 2005.
 [0002]This invention relates to a process to rapidly and automatically find endmembers of a data set made up of spectra, such as a spectral image. Such endmembers are used in a number of applications, such as material classification.
 [0003]Endmembers are spectra that are chosen to represent the most “pure” surface materials from which the pixels in a spectral image are composed. Mathematically, they are basis spectra whose physically constrained linear combinations match the pixel spectra (to within some error tolerance), but which themselves cannot be represented by such linear combinations. “Physically constrained” means constrained by positivity, at least. Endmembers that represent radiance spectra must satisfy the positivity constraint. Other physicallybased constraints may be imposed, such as sumtounity (i.e., the pixels are weighted mixtures of the endmembers) or sumtounity or less (i.e., the pixels are weighted mixtures of the endmembers plus “black”). The latter constraint is common for reflectance spectra. The invention allows selection of any of these constraints.
 [0004]There are two different categories of endmembers and several different methods and algorithms for finding them. The first category consists of endmembers that do not necessarily correspond to specific pixels in the image. For example, they may represent materials of a purer composition than occur in the scene. Such spectra might be obtained from a library of laboratorymeasured reflectances for a variety of materials that might be present. Alternatively, the endmembers may represent cluster averages, which match many spectra well but none exactly.
 [0005]The invention (sometimes termed “SMACC” herein) relates primarily to the second category of endmembers, which are actual pixels in the image. There are several wellknown algorithms for finding these endmembers. IDL's ENVI software contains a method based on a “Pixel Purity Index” [ENVI Users Guide, Research Systems, Inc., 2001] and supervised Ndimensional visualization. This method is not automated (it requires manual operation by an analyst) and is fairly timeconsuming. Another method, called NFINDR [http://www.sennacon.com/nfindr/], chooses endmembers based on a maximumsimplexvolume criterion, and is fully automated and reasonably fast.
 [0006]There are many uses of endmembers, including classification, detection and data compression. The endmembers can be used to identify unique materials in the scene, and thus can be input to classification routines. They can also be used in a constrained leastsquares unmixing routine to find targets and their pixel fill fraction, as an alternative to matched filtering. If the number of spectral channels is large, the endmember abundances are sparse (most values are zero), so the abundance image represents an efficient compression of the original data cube. Upon matching the endmembers to library materials, the abundances define the surface material composition of the scene. This enables one to estimate various physical properties, such as surface reflectance at wavelengths not originally measured.
 [0007]The inventive SMACC method is similar to NFINDR in its speed and automation. However, it uses a different mathematical criterion, termed residual minimization, for finding the endmembers, and thus produces somewhat different results. In addition, SMACC simultaneously generates estimated endmember weights (abundances) for each pixel, and, unlike NFINDR, it can be used to generate more endmembers than there are spectral channels. This may be useful for multispectral data.
 [0008]A drawback of SMACC is that it can be adversely affected by noise. When a large number of endmembers is sought, there may be redundancies (i.e., pixels identified as endmembers that actually are nearly identical to one another). In addition, the SMACC endmember weights do not match full constrained leastsquares results, but rather are stepwise constrained least squares results. However, the SMACC results, which are obtained with less computational and/or analyst time, are similar to those from other methods.
 [0009]Typical endmember algorithms are most efficient when the entire spectral image to be analyzed fits in the random access memory (RAM) of the computer processor. However, images that contain a very large number of pixels and/or spectral channels may be too large to fit in the RAM, causing the processor to spend additional time repeatedly transferring portions of the data between the RAM and the computer hard disk or other storage medium. The SMACC invention includes a method for finding endmembers for an image of arbitrary size while minimizing this additional transfer time, by dividing the image into smaller images that fit in the RAM.
 [0010]The SMACC method allows the user to select positivityonly, sumtounity, or sumtounityorless constraints on the endmember weights. The positivityonly option is appropriate for unmixing reflectance spectra under conditions of variable illumination. In this case, the sum of the endmember abundances for a given pixel may exceed unity. The sumtounityorless option is recommended when a strict physical interpretation of the abundances in terms of material and shadow fractions is desired; the results are typically similar to the positivityonly case but not identical. The sumtounity option is recommended for unmixing spectra, such as radiances or thermal IR emissivities, when a zero endmember is not physically plausible, or when it is desired to find very dark endmembers such as shadow endmembers. The second endmember found is among the darkest pixels in the scene, if not the darkest.
 [0011]This invention features a process for determining a subset of members of a group of N data vectors, such as spectra, the subset denoted as endmembers, which may be taken in positive linear combinations to approximate the other members of the group, comprising a. providing a data set (for example an image data) comprised of a plurality of spectra (for example, pixels in which each pixel comprises a spectrum), b. defining an error metric dependent on a difference between two spectra, c. selecting a first spectrum as the first endmember, d. for the member spectra in the group, determining a nonnegative weighting factor such that, when the first spectrum is multiplied by the weighting factor and subtracted from the spectrum, the resulting difference generates a smallest error metric, e. selecting as a next endmember the member spectrum whose calculated difference from step d generates the largest error metric, f. for the N member spectra in the group, determining a weighting factor, determining an updated difference by subtracting from the prior difference the difference for the endmember in step e multiplied by the weighting factor, and determining an updated set of member weighting factors by subtracting from the prior set of weighting factors the set of weighting factors for the endmember in step d multiplied by the weighting factor, such that all updated weighting factors are nonnegative and the updated member difference generates a smallest error metric, and g. repeating steps e and f for all or a subset of member spectra one or more times.
 [0012]The process may further comprise multiplying each endmember by the weighting factor, to create a weighted endmember value. The process may further comprise approximating each member of a group of N spectra by means of a sum of the weighted endmember values. The invention can also feature a process for determining a subset of members of a group of N spectra, the subset denoted as endmembers, which may be taken in positive linear combinations to approximate the other members of the group, comprising dividing the group of N spectra into subgroups, determining endmembers of each subgroup by the process described above, forming the endmembers of the subgroups into an endmember group, and determining endmembers of the endmember group by the process described above. The group of N spectra may be divided into subgroups, and the weighting factors for the members in each subgroup may be determined.
 [0013]The error metric may be the mean squared spectrum difference. The first spectrum may include the spectrum in the group with the largest mean or mean squared value. The first spectrum may include a target spectrum in the group. The first spectrum may include a spectrum that is not in the group.
 [0014]The determinations may include the additional condition that the sum of the weighting factors does not exceed one, or the sum of the weighting factors may equal one. Step g above may be carried out for only those member spectra for which the differences from step d are greater than a predetermined value. Step g may be carried out for all member spectra until the largest difference from step d among all the member spectra is less than a predetermined value. In the case in which the data set comprises image data, if a sensor is used to gather the image data, the predetermined value may be estimated from the sensor noise. Step g may be terminated when a predetermined number of endmembers have been selected. The difference between one and the sum of the weighting factors is output as a shade weighting factor.
 [0015]This invention also features a process for approximating each member of a group of N data vectors, such as spectra, by means of a positive linear combination of a prior selected subset of members, the subset denoted as endmembers, each endmember being multiplied by a corresponding positive weighting factor, in which the process for determining the weighting factors comprises a. providing a data set (for example an image data) comprised of a plurality of spectra (for example pixels in which each pixel comprises a spectrum), b. defining an error metric dependent on a spectrum difference, c. selecting a first endmember as a starting spectrum, d. for each of the N member spectra in the group that is not the starting spectrum, determining a nonnegative member weighting factor such that, when the starting spectrum is multiplied by the member weighting factor and subtracted from the member spectrum, the resulting member difference generates a smallest error metric, e. selecting a next endmember, f. for each of the N member spectra in the group that is not the starting spectrum or any selected endmember, determining a weighting factor, determining an updated member difference by subtracting from the prior member difference the difference for the endmember in step e multiplied by the weighting factor, and determining an updated set of member weighting factors by subtracting from the prior set of member weighting factors the set of weighting factors for the endmember in step e multiplied by the weighting factor, such that all updated member and endmember weighting factors are nonnegative and the updated member difference generates a smallest error metric, and g. repeating steps e and f until all endmembers have been selected.
 [0016]In this process, a subset of members of a group of N spectra may be determined, the subset denoted as endmembers, which may be taken in positive linear combinations to approximate the other members of the group. This may be accomplished by dividing the group of N spectra into subgroups, determining endmembers of each subgroup by the process, forming the endmembers of the subgroups into an endmember group, and determining endmembers of the endmember group by the process. The group of N spectra may be divided into subgroups, and the weighting factors for the members in each subgroup may be determined.
 [0017]Other objects, features and advantages will occur to those skilled in the art from the following description of the invention and its preferred embodiments, including the FIGURE, which is a flow chart of the process of the two preferred embodiments described below.
 [0000]Nomenclature and Definitions
 [0018]For ease of explanation, let the set of data vectors (which must be of equal length) be a group of N spectra (in the preferred embodiment the spectra are pixel spectra), in which each pixel spectrum consists of intensity values for a set of spectral channels.
 [0019]The set of pixel spectra is denoted {r_{1}r_{2 }. . . r_{N}}. An arbitrary individual member of this set is denoted r*.
 [0020]The endmembers to be determined are a subset of the pixel spectra and are denoted {e_{1 }e_{2 }. . . e_{M}}, where M is the number of endmembers. In addition, e_{0 }is defined as the spectrum consisting of all zero intensities, referred to as the zero endmember; it is used as a placeholder to implement an optional summation constraint as will be described.
 [0021]Let R*=the pixel spectrum vector difference (also referred to as the residual) between r* and its representation via the expression Σ_{k}A*_{k }e_{k}, where k runs from 1 to M. A*_{k }is referred to as the weight (or weighting factor) of endmember k in the r* spectrum. In the preferred embodiments, A*_{k }is also defined for k=0, although its value does not affect the residual. A*_{0 }is set to 1 in the first preferred embodiment and 0 in the second preferred embodiment.
 [0022]When r* is an endmember, e_{j}, its residual is R_{j}. The representation of e_{j }is written as Σ_{k}A_{jk }e_{k}, where k≠j. A_{jk }is referred to as the weight (or weighting factor) of endmember k in endmember j.
 [0023]Let x denote the absolute value (length) of a spectrum vector x.
 [0024]Let p(x,y) denote the projection length of spectrum vector x onto spectrum vector y, given by x·y/y.
 [0000]Algorithm Description for the First Preferred Embodiment
 [0025]This first preferred embodiment algorithm provides a positivity constraint on the weights, and, optionally, a constraint that the weights for each pixel spectrum must sum to unity or less. If the latter constraint is not used, then the zero endmember variables (i.e., A*_{0}) are not needed; however, for ease of explanation they are retained in both cases in the following description.
 [0000]Step a: Define an Error Metric
 [0026]Define the error metric as the mean square difference between two spectra (the mean of the squares of the differences of the corresponding channels).
 [0000]Step b: Select a Starting Spectrum as the First Endmember
 [0027]Set the weights of the 0^{th }endmember to unity, i.e., A*_{0}=1, and select the pixel spectrum with the largest mean intensity squared or the largest mean intensity absolute value as endmember 1 (e_{1}).
 [0000]Step c: For Each Pixel Spectrum, Determine the Weighting Factor for the First Endmember
 [0028]The weighting factor, A*_{1}, is given by the larger of zero and p(r*,e_{1})/e_{1}. If it is desired to impose the optional constraint that the weights for each pixel sum to unity or less, A*_{1 }is restricted to be unity or less; i.e., A*_{1}≦1. With this weighting factor definition, it can be shown that the mean square of the residual
R*=r*−A* _{1} e _{1} (1a)
(i.e., the error metric) is minimized with respect to the allowable values of A*_{1}. This residual represents the difference between the pixel spectrum and its representation by the first endmember alone.  [0029]Next, the j=0 endmember weights A*_{0 }are updated by subtracting A*_{1}; i.e., the updated weights are A*_{0}=1−A*_{1}.
 [0030]The following steps d and e constitute an iteration, or cycle, which is repeated as described in step f.
 [0000]Step d: Select the Next Endmember
 [0031]Select as the next endmember the pixel spectrum for which the most recently calculated residual R* (from the previous iteration or from step c) yields the largest error metric.
 [0000]Step e: For Each Pixel Spectrum, Determine the Weighting Factor for the New Endmember Found in Step d.
 [0032]For ease of illustration, the method is described below for endmember k, taken as an example.
 [0000]Step e1: Calculate Provisional Weighting Factors by Projection.
 [0033]The provisional values of A*_{k }are larger of zero and p(R*,R_{k})/R_{k}. The provisional updated residuals are
R*=R* ^{prev} −A* _{k} R _{k.}, (1b)
where R*^{prev }are the most recently calculated residuals (from the previous iteration or from step c). This residual is equal to the difference between the pixel spectrum and its representation by a weighted sum of the endmembers determined up to this point. If it is desired to impose the optional constraint that the weights for each pixel must sum to unity or less, the provisional A*_{k }is restricted to be unity or less (i.e., A*_{k}≦1).  [0034]In the updated pixel spectrum representation, the new endmember replaces a combination of prior endmembers that were used to represent it. Therefore, the previously determined weights of the prior endmembers in the pixel spectra no longer hold and must be updated; this is done in Step e2 below. In the updating process, the weights must not be allowed to become negative. This places an upper limit on the allowable value of the new endmember weight A*_{k}, which is why the value calculated in step 1 is termed “provisional.” The A*_{k }value accounting for this upper limit is calculated in step e2, and the updated weights of the prior endmembers are calculated in step e4.
 [0000]Step e2: Find Upper Limits of New Endmember Weights
 [0035]For the previously determined weights to remain nonnegative upon updating (step e4), an upper limit on A*_{k }is determined. To ensure that after A*_{k}A_{jk }is subtracted A*_{j }remains nonnegative, the inequality A*_{j} ^{prev}≧A*_{k}A_{jk }must be satisfied. For a given previous endmember j>0, the maximum permissible value of A*_{k }is given by the ratio A*_{j} ^{prev}/A_{jk}; therefore, the smallest of these ratios for all j>0 is found, and this ratio is set as the A*_{k }upper limit value. The A*_{k }value is then reset to be the smaller of the A*_{k }upper limit value and the A*_{k }provisional value given in Step e1.
 [0036]For a given pixel, the sum over all j of the endmember weights Σ_{j}A*_{j }remains at unity from each iteration to the next. Therefore, if it is chosen to extend the inequality condition A*_{j} ^{prev}≧A*_{k}A_{jk }to j=0, thereby imposing a nonnegativity constraint on A*_{0}, the sum Σ_{j}A*_{j }is constrained to be unity or less over j>0 (i.e., over the nonzero endmembers). If this is not chosen, the sum is allowed to exceed unity.
 [0000]Step e3: Update the Spectral Residuals and Error Metrics
 [0037]The pixel and endmember residuals are updated via Eq. (1b) using the A*_{k }value determined from Step e2. The error metrics are recalculated using the updated residuals.
 [0000]Step e4: Update the Weights
 [0038]The weights A*_{j }(where j<k) are updated via
A* _{j} =A* _{j} ^{prev} −A* _{k} A _{jk}. (2)
Step f: Repeat Steps d and e  [0039]Repeat steps d and e for endmembers 3, 4, etc., until the desired number of endmembers has been reached (i.e., k=M) and/or the error metrics have been reduced to smaller than a desired tolerance. For example, a tolerance may placed on the individual pixel error metrics, such that the repetition of steps d and e is halted for those pixels that have a smaller error metric; the repetition continues for those pixels that have a larger error metric. Alternatively, a tolerance may be placed on the largest pixel error metric, such that the repetition of steps d and e is halted for all pixels when they all have an error metric smaller than the tolerance.
 [0000]Algorithm Description for the Second Preferred Embodiment
 [0040]This second preferred embodiment algorithm provides a strict sumtounity constraint on the weights for each pixel. It is identical to the first preferred embodiment algorithm except that:

 1. the starting values of A*_{0 }are set to 0 in step b,
 2. in step c, the A*_{1 }are set to 1, and
 3. in step e2, the weight of the j=0 endmember is included in the determination of the upper limit, i.e. A*_{j} ^{prev}≧A*_{k}A_{jk }for j=0 to k−1.

 [0044]By applying the inequality (nonnegativity condition) in step e.2 to j=0, A*_{0 }remains zero. In combination with the sumtounity constraint on the A*_{j }where j=0 is included, this forces Σ_{j}A*_{j}=1 for j>0.
 [0000]Extension of the Preferred Embodiments to Arbitrarily Large Images
 [0045]For efficiently determining the endmembers of an image that is too large to fit into the computer RAM, the following method may be used with either the first or second preferred embodiment algorithms:

 divide the image consisting of a group of N spectra into subgroups of spectra;
 determine endmembers of each subgroup by the preferred embodiment algorithm;
 form the endmembers of the subgroup into an endmember group;
 by the preferred embodiment algorithm, determine endmembers of the endmember group, which constitute endmembers of the image.
Applications to Other Data Sets

 [0050]The invention applies to data sets comprised of a plurality of spectra. The data vectors are typically, but not necessarily, spectra from images. Alternatively, the data vectors can be temporal, such as a time sequence of spectra.
 [0051]Other details may be set forth in the provisional patent application from which priority is claimed, the entire disclosure of which is incorporated herein by reference.
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